CALIFORNIA STATE UNIVERSITY, SACRAMENTO
Department of Mechanical Engineering
Paskowitz Spring 2000
Course Web Page http://webpages.csus.edu/~passki/
Spreadsheet Exercise 1
Calculating Ionic Bonding Forces
ENG-45
Introduction
Spreadsheets are valuable tools for any student or practicing
engineer who does calculations. They can aid in performing complex
calculations and graphing the results, for database applications and
for importing data from experiments for analysis and graphing. Modern
spreadsheets provide a number of advanced numerical tools which
allows one to solve complex equations. Oddly, many students and even
practicing engineers don't know how to use a spreadsheet. Therefore
we offer this exercise to put these valuable tools in your hands to
help you solve many of the problems presented during this course and
in future courses you take.
This exercise involves generating a spreadsheet which will produce
a graph that illustrates a fundamental concept in materials science.
It utilizes three equations and their associated parameters plus
several physical constants. We approach this calculation in a simple,
progressive manner which will allow one to examine each aspect of the
program as it is added and to build upon its basic foundation. The
approach taken here is a good basic approach to building any reusable
spreadsheet program, especially those which will continue to
grow.
The Problem: Ionic Bonding
Potentials
The forces involved in ionic bonding are described in section 2.5 of Van Valck's text. Briefly, the force holding two oppositely charged ions at a particular distance from each other consists of two parts. The first is the force of the coulombic attraction between the oppositely charged ions
where ko is a constant, a is the interatomic spacing, Z is the
valence and q the charge (±1.602xl0-19 C) on each
ion. The second part is the repulsion by the overlapping of
negatively charged electric fields and the positively charged
nuclei
where, l and r
are both constants. The total bonding force is equal to the
sum of the repulsive and attractive forces.
The General Layout of the
Spreadsheet
Define the Constants and Parameters Used by the Equations
All parameters used by the equations above should be defined in the spreadsheet independently of the equations. While one could enter them as constants in the cells containing the equations they should occupy cells in another part of the spreadsheet. The reason for this is that we'd like to be able to examine and experiment with the values of these parameters.
The parameters used by these equations are actually functions of temperature. Instead of entering the value for these constants, we could enter the temperature dependent function instead and define the temperature in yet another cell. That way we could experiment with the effect of temperature on ionic bonding. While this would be a much more interesting calculation we would not have to alter any other part of the spreadsheet. The equations for the interatomic forces would still reference the same cells to obtain the values of the parameters.
In general, it is better to define the parameters used by an
equation and other constants such as physical constants in cells
rather than in the equations themselves. This makes it easier to
debug the program, to change the units and to experiment with the
equations.
Enter the Equations for the Interatomic Forces in Columns
The main part of the spreadsheet will be a series of columns of
equations for the independent and dependent variables. In this
exercise the independent variable is the interatomic spacing while
the dependent variables are the forces of attraction, repulsion and
their sum which is the total bonding force. The dependent variables
could be combined into one large equation, and one column, but long
equations are difficult to debug so it is best to keep them simple.
Plus we'd like to plot each force to see how they relate to the total
bonding force.
Construct a Graph
The graph will be a standard x-y graph with interatomic spacing as
the x axis and the interatomic forces plotted along the y axis. The
line colors and styles plus the legend will be used to identify the
different forces.
Construct the Spreadsheet
Header
Start by entering the basic details about this spreadsheet program
such as the title, owner and date it was created. Next, enter a brief
description of this calculation and any literature citations that
might be important. This header can be very useful when referring to
this work at a later date.
Parameters
Enter the values for each parameter for each equation in cells
just below the spreadsheet's header. Include the names of the
parameters and the units you are using. If the units are not those
you will need to use in the equations then include a cell with the
appropriate units. This cell should be a simple calculation using a
units conversion factor. Your equations will reference these cells,
saving you the trouble of including the conversion factors in the
equations.
The Independent Variable - Interatomic Spacing
Put the interatomic spacing in the first column. Start with a
value of 0 at the top of the column and in the next cell add to it
the increment da. Repeat this for the next 300 or so cells in the
column, referencing the cell above for the value to add da to. For
maximum flexibility the increment da should be defined in a separate
cell in the parameters section of this spreadsheet. This will let you
easily change the range and resolution of your calculations.
The Dependent Variables - Attractive and Repulsive Bonding Forces
In adjacent columns enter the equations for the attractive and
repulsive forces. Enter it at the top of the column and replicate it
so that it uses each of the cells in first column (interatomic
spacing).
Total Bonding Force
Add columns 2 and 3 together in column 4.
Annotate
Label each column. Don't forget to include the units.
Graph
Set up an xy graph where column 1 contains the values for the x
axis and columns 2 through 4 contain the values for the first 3
series of the y axis. Label the axes appropriately and include a
legend. Insert the graph into the spreadsheet, above the columns and
to the right of the parameters.
Print the graph and also the top portion of the spreadsheet which
includes the parameters, results and the inserted graph.
Further Analysis
This basic spreadsheet can be used to calculate the equilibrium spacing between the ions, Young's modulus and the bulk modulus. One can even calculate the energy of interatomic bonding and then estimate the melting point of the material.
1 . Examine the column containing the total bonding force and find the value of the interatomic spacing where the force is equal to zero. This is the equilibrium interatomic spacing of the ions.
3. Calculate the bulk modulus using the equation
where n is the number of atoms in the unit cell. Compare this to the bulk modulus calculated using
4. Calculate the nominal strain this material would experience if stressed to its yield point, 100 MPa.
5 . The bonding energies are related to the attractive, repulsive and total bonding forces by
Calculate these bonding energies and plot the results.
7. What is the bonding energy? Is it a reasonable value?
8. Estimate the melting point of this material. (Assume that
these calculations refer to conditions at 0 K.)
Special Thanks to Dr. Mike Meier from UC Davis for providing this material for the lab section.
© Dr. Michael L. Meier 1999